Periodic Shock Waves Research Articles

Particle drift in a resonance tube--a numerical study.

The paper considers longitudinal drift of small particles in a resonance tube, caused by periodic shock waves, and its effect on particle agglomeration. It is found that depending on particle size, drift is caused by shock waves and/or gas acceleration and compression. It is also shown that the drift velocity and direction can be controlled by the frequency of the piston that causes gas oscillations in the resonance tube. The obtained numerical solutions indicate that particle drift in a resonance tube enhances aerosol agglomeration. An agglomeration kernel is derived for this case, accounting for particle drift, leading to an estimate of agglomeration time. The time predicted by present model is of the same order of magnitude as that obtained from experiments in the literature.

Radial pulsations of helium stars with masses from 1 to 10 M ⊙

We present the results of our hydrodynamic calculations of radial pulsations in helium stars with masses 1 M⊙ ≤ M ≤ 10 M⊙, luminosity-to-mass ratios 1 × 103L⊙/M⊙ ≤ L/M ≤ 2 × 104L⊙/M⊙, and effective temperatures 2 × 104 K ≤ Teff ≤ 105 K for mass fractions of helium Y=0.98 and heavy elements Z=0.02. We show that the lower boundary of the pulsation-instability region corresponds to L/M ∼ 103L⊙/M⊙ and that the instability region for L/M ≲ 5 × 103L⊙/M⊙ is bounded by effective temperatures Teff ≲ 3 × 104 K. As the luminosity rises, the instability boundary moves into the left part of the Hertzsprung-Russell diagram and radial pulsations can arise in stars with effective temperatures Teff ≲ 105 K at L/M ≳ 7 × 103L⊙/M⊙. The velocity amplitude for the outer boundary of the hydrodynamic model increases with L/M and lies within the range 200 ≲ ΔU ≲ 700 km s−1 for the models under consideration. The periodic shock waves that accompany radial pulsations cause a significant change of the gas-density distribution in the stellar atmosphere, which is described by a dynamic scale height comparable to the stellar radius. The dynamic instability boundary that corresponds to the separation of the outer stellar atmospheric layers at a superparabolic velocity is roughly determined by a luminosity-to-mass ratio L/M ∼ 3 × 104L⊙/M⊙.

On the shock-induced variability of emission lines in M-type Mira variables

Our observations of cool, shock penetrated, expanding atmospheres of M-type Mira variables (Richter & Wood 2001) have suggested that the emission lines of FeII and (FeII) can serve as an excellent diagnostic tool to study the hydro- and thermodynamical conditions in the shocked regions close to the photosphere of these stars. Here we present a series of detailed NLTE (Non Local Thermodynamic Equilibrium) radiative transfer calculations, which have been performed on structures re- sulting from thermodynamical models of periodic shock waves, in order to calculate the emergent FeII and (FeII) emission line fluxes and to analyse the conditions which lead to their formation. Our basic parameter studies reveal that the ionised iron lines originate from the hot post-shock zone and that they are in fact emitted close to the star's photosphere. Furthermore, the modelling of the FeII and (FeII) emission line fluxes determine a specified limit of the shock velocity amplitude and pre-shock density for the innermost shocks. This offers an unique possibility to determine the thermodynamical conditions in the inner dust formation zone and thereby will shed some light on the basic mechanism of dust formation in M-type Mira stars.

An electrostatic model for nonlinear waves in the upper ionosphere

According to observations by the Freja satellite in the upper ionosphere, amplitudes of magnetic disturbances associated with observed density solitons are only several 10 −2 nT or less. These very weak magnetic disturbances can be neglected against the ambient magnetic field of 0.2 Gauss in Freja's orbit region. In this paper we establish an electrostatic model for the nonlinear waves observed on board of Freja. The fluid equations are used and the “Sagdeev potential” is derived to study the nonlinear waves. The results show the existence not only of solitons with a density hump and solitons with a density dip, but also of periodic density waves and density shock waves. The theoretic amplitudes of the solitons may vary between 20% and a factor of 2, which is consistent with the observations on board of Freja. Some waveforms of the solitons with a density dip are very similar to the observed ones.

Spectrum of shock waves produced by an optical discharge at a high laser-pulse repetition rate

The acceleration of a plasma jet by an optical pulsating discharge is studied. The optical discharge generated periodic shock waves or shock waves combined in periodic trains. For a pulse repetition rate f > 50 kHz, the line at frequency f dominated in the spectrum of periodic waves. The spectrum of trains following with the frequency F0 ≪ f contained an ultrasonic component and a low-frequency component. The trains produces a strong acoustic effect. The average power of the waves amounted to 160 W, and the conversion efficiency of laser radiation to shock waves was ~10% — 25%.

Agglomeration of submicrometer particles in weak periodic shock waves

Agglomeration of submicrometer particles in the presence of acoustic fields was treated experimentally. Fast agglomeration of aerosol particles in weak periodic shock waves was observed. The rate of particle agglomeration in shock waves was found to be much faster than in continuous sound waves.

Effects of weak periodic pressure waves on time to ignition of fuel mixtures

The accumulation of small-amplitude gas dynamic perturbations accelerates the process of self-ignition of a homogeneous explosive mixture. In the present paper the cumulative effect of forced acoustic oscillations is studied when timescales of piston motions in a closed cylinder are comparable with the acoustic timescale. It is shown that the most significant shortening of the time to ignition takes place in a resonant system, where periodic shock waves travel back and forth in the cylinder.

TRANSVERSE RESONANT OSCILLATIONS IN ACOUSTIC DUCTS

A theoretical study is made of the flow induced by a piston, performing rotational sinusoidal vibrations in a uniform two-dimensional planar duct. A frequency of the forced vibrations is close to a cut-off frequency. It is shown that exactly at a cut-off frequency the steady-state response represents a transverse periodic shock wave, propagating between duct walls and decaying algebraically in the far field.

Solitary shock waves and other travelling waves in a general compressible hyperelastic rod

In the literature, it has been conjectured that solitary shock waves can arise in incompressible hyperelastic rods. Recently, it has been shown that this conjecture is true. One might guess that when compressibility is taken into account, such a wave, which is both a solitary wave and a shock wave, can still arise. One of the aims of this paper is to show the existence of this interesting type of wave in general compressible hyperelastic rods and provide an analytical description. It is difficult to directly tackle the fully nonlinear rod equations. Here, by using a non–dimensionalization process and the reductive perturbation technique, we derive a new type of nonlinear dispersive equation as the model equation. We then focus on the travelling–wave solutions of this new equation. As a result, we obtain a system of ordinary differential equations. An important feature of this system is that there is a vertical singular line in the phase plane, which leads to the appearance of shock waves. By considering the equilibrium points and their relative positions to the singular line, we are able to determine all qualitatively different phase planes. Those paths in phase planes which represent physically acceptable solutions are discussed one by one. It turns out that there is a variety of travelling waves, including solitary shock waves, solitary waves, periodic shock waves, etc. Analytical expressions for all these waves are obtained. A new phenomenon is also found: a solitary wave can suddenly change into a periodic wave (with finite period). In dynamical systems, this represents a homoclinic orbit suddenly changing into a closed orbit. To the authors9 knowledge, such a bifurcation has not been found in any other dynamical systems.

Turbulent acoustic streaming excited by resonant gas oscillation with periodic shock waves in a closed tube

Resonant gas oscillations with periodic shock waves in a closed tube are studied by executing large-scale computations of the compressible 2-D Navier–Stokes equations with a finite-difference method. In a quasisteady state of oscillation, acoustic streaming (mean mass flow) of large Rs is excited, where Rs is the streaming Reynolds number based on a characteristic streaming velocity, the tube length, and the kinematic viscosity. When Rs=560, relatively strong vortices are localized near the tube wall. The resulting streaming pattern is almost stationary but quite different from that of the Rayleigh streaming. The streaming of Rs=6200 involves unsteady vortices in a region near the center of the tube. Turbulent streaming appears in the result of Rs=56000, where vortices of various scales are irregularly generated throughout the tube.

Particle motion in resonance tubes

Small particle motions in standing or travelling acoustic waves are well known and extensively studied. Particle motion in weak shock waves has been studied much less, especially particle motion in periodic weak shock waves which as yet has not been dealt with.The present study considers small particle motions caused by weak periodic shock waves in resonance tubes filled with air. A simple mathematical model is developed for resonance gas oscillations under the influence of a vibrating piston with a finite amplitude at the first acoustic resonance frequency. It is shown that a symmetrical sinusoidal piston motion generates non-symmetric periodic shock waves. A model of particle motion in such a flow field is suggested. It is found that non-symmetric shock waves cause particle drift from the middle cross-section toward the ends of the resonance tube. The velocity of particle drift is found to be of the order of Dpρp/ Trρg, where Dp is the particle diameter, Tr the period of the resonance oscillation, ρp and ρg are the particle and gas density, respectively. At the same time, the velocity drift strongly depends on the ratio τ/Tr, where τ is the particle relaxation time. Particle drift is vigorous when τ/Tr∼1 and insignificant when τ/Tr 1. Theoretical predictions of particle drift in resonance tubes are verified numerically as well as experimentally.When the particle relaxation time is much smaller than period of the resonance oscillations particles perform oscillations around their equilibrium positions with amplitude of the order of Dpρp/ρg. It is shown that the difference in oscillation amplitude of particle of difference sizes explains coalescence of aerosol droplets observed in experiments of Temkin (1970).The importance of the phenomena for particle separation, coagulation and transport processes is discussed.

Resonance gas oscillations in closed tubes

The problem of gas motion in a tube closed at one end and driven at the other by an oscillating poston is studied theoretically. When the piston vibrates with a finite amplitude at the first acoustic resonance frequency, periodic shock waves are generated, travelling back and forth in the tube. A perturbation method, based on a small Mach number. M and a global mass conservation condition, is employed to formulate a solution of the problem in the form of two standing waves separated by a jump (shock front). By expanding the equations of motion in a series of a small parameter ε = M½, all hydrodynamic properties are predicted with an accuracy to second-order terms, i.e. to ε2. It is found that the first-order solution coincides with the previous theories of Betchov (1958) and Chester (1964), while additional terms predict a non-homogeneous time-averaged pressure along the tube. This prediction compares favourably with experimental results from the literature. The importance of the phenomenon is discussed in relation to different transport processes in resonance tubes.

Periodic shock waves in spherical resonators (survey)

This article surveys the literature on the problem of shock waves in spherical resonators. The published data is used to examine the feasibility of exciting shock waves in such resonators by means of a source of low-amplitude harmonic oscillations. A nonlinear wave equation is obtained to describe the propagation of unidimensional spherical waves in solids, liquids, and gases, as well as in bubbly liquids. A solution to the equation is constructed by the small-parameter method with the use of traveling-wave functions. Then, in solving boundary-value problems, linearized equations are integrated and the resonance frequencies at which the amplitudes of the oscillation increase without limit according to the linear solution are determined. Near these frequencies, the linear analysis is then refined by allowing for the nonlinear terms in the boundary-value problems. It is shown that an increase in the amplitude of the oscillations at resonance frequencies may lead to the formation of spherical periodic shock waves in the given resonators. An analogy is made between these waves and resonance shock waves excited in long unidimensional resonators.

Resonant wave motion in a non-homogeneous system

A nonlinear theory of resonant wave motion in an inhomogeneous system (arising from different mechanical applications) is considered. Depending on the magnitude of the influence of inhomogeneity two different situations are encountered. The system may behave either as a nonlinear one-degree-of-freedom oscillator exhibiting a response curve with either soft or hard spring behaviour or for a sufficiently small influence of inhomogeneity, there are periodic shock waves in a certain frequency band about the linear resonance frequency, a phenomenon that is familiar from homogeneous systems like a gas filled tube being excited close to resonance.

Exact solutions of a hierarchy of mixing speeds models

This paper presents several new aspects of discrete kinetic theory (DKT). First a hierarchy of d-dimensional (d=1,2,3) models is proposed with (2d+3) velocities and three moduli speeds: 0, 2, and a third one that can be arbitrary. It is assumed that the particles at rest have an internal energy which, for microscopic collisions, supplies for the loss of the kinetic energy. In a more general way than usual, collisions are allowed that mix particles with different speeds. Second, for the (1+1)-dimensional restriction of the systems of PDE for these models which have two independent quadratic collision terms we construct different exact solutions. The usual types of exact solutions are studied: periodic solutions and shock wave solutions obtained from the standard linearization of the scalar Riccati equations called Riccatian shock waves. Then other types of solutions of the coupled Riccati equations are found called non-Riccatian shock waves and they are compared with the previous ones. The main new result is that, between the upstream and downstream states, these new solutions are not necessarily monotonous. Further, for the shock problem, a two-dimensional dynamical system of ODE is solved numerically with limit values corresponding to the upstream and downstream states. As a by-product of this study two new linearizations for the Riccati coupled equations with two functions are proposed.

Exact solutions for a higher-order nonlinear Schrödinger equation.

We performed a systematic analysis of exact solutions for the higher-order nonlinear Schr\"odinger equation i${\mathrm{\ensuremath{\psi}}}_{\mathit{X}}$+${\mathrm{\ensuremath{\psi}}}_{\mathit{T}\mathit{T}}$=${\mathit{a}}_{1}$\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\psi}${\mathrm{\ensuremath{\Vert}}}^{2}$+${\mathit{a}}_{2}$\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\psi}${\mathrm{\ensuremath{\Vert}}}^{4}$+${\mathit{ia}}_{3}$ (\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\psi}${\mathrm{\ensuremath{\Vert}}}^{2}$${)}_{\mathit{T}}$+(${\mathit{a}}_{4}$+${\mathit{ia}}_{5}$)\ensuremath{\psi}(\ensuremath{\Vert}\ensuremath{\psi}${\mathrm{\ensuremath{\Vert}}}^{2}$${)}_{\mathit{T}}$ that describes wave propagation in nonlinear dispersive media. The method consists of the determination of all transformations that reduce the equation to ordinary differential equations that are solved whenever possible. All obtained solutions fall into one of the following categories: ``bright'' or ``dark'' solitary waves, solitonic waves, regular and singular periodic waves, shock waves, accelerating waves, and self-similar solutions. They are expressed in terms of simple functions except for few cases given in terms of the less-known Painlev\'e transcendents.

Interaction of air shock waves with porous compressible materials

J. J. Keller, "Nonlinear acoustic resonances in shock tubes with varying cross-sectional area," Z. Angew. Math. Phys., 28, No. i, 107 (1977). L. P. Gor'kov, "Nonlinear acoustic oscillations of a gas column in a closed tube," Inzh. Zh., 3, No. 2 (1963). W. Chester, "Resonant oscillations in closed tubes," J. Fluid Mech., 18, Part ii (1964). A. N. Kraiko and A. L. Ni, "Nonlinear acoustics approximation in problems of gas oscillations in tubes," Prikl. Mat. Mekh., 44, No. 1 (1980). A. L. Ni, "Nonlinear resonant oscillations of a gas in a tube under the action of a periodically varying pressure," Prikl. Mat. Mekh., 47, No. 4 (1983). Sh. U. Galiev, N. A. Ii'gamov and A. V. Sadykov, "Periodic shock waves in a gas," Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2 (1970). M. P. Mortell and B. R. Seymour, "A finite-rate theory of quadratic resonance in a closed tube," J. Fluid Mech., 112, 411 (1981).

Polarization properties and time variations of the SiO maser emission of R Leo

view Abstract Citations (31) References (24) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Polarization properties and time variations of the SiO maser emission of R Leonis. Clark, F. O. ; Troland, T. H. ; Pepper, G. H. ; Johnson, D. R. Abstract Measurements over a period of 3.5 years of the polarization properties of the v = 1, J = 2-1 SiO circumstellar emission from R Leo have yielded line profiles which are well represented by a superposition of narrow Gaussian features, each having a well defined position angle of maximum linear polarization, which persist over an optical period and then disappear at or near optical maximum, perhaps due to the passage of periodic atmospheric shock waves. The continuity observed in the Gaussians identifies an element of order in the SiO masers for the first time. On the basis of the present data, it is suggested that the SiO maser is located very close to the star, and that the dominant gain path is tangent to the stellar surface. A transient event may have been witnessed which temporarily increased the mass of material in the SiO-emitting region, and accelerated the new material as well, perhaps by means of radiation pressure on newly formed molecules in the maser region. Publication: The Astrophysical Journal Pub Date: January 1984 DOI: 10.1086/161646 Bibcode: 1984ApJ...276..572C Keywords: Interstellar Masers; Polarization Characteristics; Stellar Envelopes; Variable Stars; Linear Polarization; Normal Density Functions; Radial Velocity; Radiant Flux Density; Silicon Oxides; Spectral Line Width; Stellar Spectra; Vibrational Spectra; Astrophysics full text sources ADS | data products SIMBAD (1)

Winds in late-type stars: Mechanisms of mass outflow

AbstractFour basic mechanisms have been proposed to explain the acceleration of winds in late-type stars –– thermal pressure gradients, radiation pressure on circumstellar dust grains, momentum addition by Alfvén waves, and momentum addition by periodic shock waves. In this review I describe recent work in applying these mechanisms to stars, and consider whether these mechanisms can work even in principle and whether they are consistent with recent ultraviolet and X-ray data from the IUE and Einstein spacecraft. Thermally-driven winds are likely important for late-type dwarfs, where the mass loss rates are small, and perhaps also in G giants and supergiants, but they cannot operate alone in the K and M giants and supergiants. Radiatively-driven winds are probably unimportant for all cool stars, even the M supergiants with dusty circumstellar envelopes. In principle, Alfvén waves can accelerate winds to high speeds provided the field lines are initially open or forced open by some mechanism, but detailed calculations are needed. Magnetic reconnection is an interesting suggestion for an acceleration mechanism when the field lines are initially closed. For the Mras and semiregular variable supergiants, periodic shock waves provide a simple way of producing rapid mass loss. Thus we are making some progress in understanding mass loss mechanisms for the cool half of the H-R diagram.

Shock wave interpretation of emission lines in long period variable stars. II - Periodicity and mass loss

An analytical description of strictly periodic shock waves passing through a stellar atmosphere is developed which allows predictions to be made of the onset of instability of the system against mass loss by hydrodynamic ejection. This diagnostic method for determining when shock-driven mass loss may be expected is compared to several numerical isothermal hydrodynamical models. The predictions by the analytical theory of the onset of mass loss are in good agreement with the numerical hydrodynamical models. The role of random aperiodicities in enhancing mass loss is investigated in the numerical models and is found to be minor. Effects of atmospheric density gradients and postshock heating are also investigated numerically and are found to be critical. We conclude that the observed mass loss rates for the long-period variables can be produced by the shock wave mechanism alone if proper account is taken of the high pressures in the hot postshock hydrogen recombination zone.